# Understanding Uncertainty, Revised Edition

## Books Praise for the First Edition

"...a reference for everyone who is interested in knowing and handling uncertainty."
Journal of Applied Statistics

The critically acclaimed First Edition of Understanding Uncertainty provided a study of uncertainty addressed to scholars in all fields, showing that uncertainty could be measured by probability, and that probability obeyed three basic rules that enabled uncertainty to be handled sensibly in everyday life. These ideas were extended to embrace the scientific method and to show how decisions, containing an uncertain element, could be rationally made.

Featuring new material, the Revised Edition remains the go-to guide for uncertainty and decision making, providing further applications at an accessible level including:

• A critical study of transitivity, a basic concept in probability
• A discussion of how the failure of the financial sector to use the proper approach to uncertainty may have contributed to the recent recession
• A consideration of betting, showing that a bookmaker's odds are not expressions of probability
• Applications of the book’s thesis to statistics
• A demonstration that some techniques currently popular in statistics, like significance tests, may be unsound, even seriously misleading, because they violate the rules of probability

Understanding Uncertainty, Revised Edition is ideal for students studying probability or statistics and for anyone interested in one of the most fascinating and vibrant fields of study in contemporary science and mathematics.

Preface xi

Prologue xiii

1. Uncertainty 1

1.1. Introduction 1

1.2. Examples 2

1.3. Suppression of Uncertainty 7

1.4. The Removal of Uncertainty 8

1.5. The Uses of Uncertainty 9

1.6. The Calculus of Uncertainty 11

1.7. Beliefs 12

1.8. Decision Analysis 13

2. Stylistic Questions 15

2.1. Reason 15

2.2. Unreason 17

Literature 17

Politics 18

Law 18

Television 18

2.3. Facts 19

2.4. Emotion 19

2.5. Prescriptive and Descriptive Approaches 20

2.6. Simplicity 22

2.7. Mathematics 23

2.8. Writing 25

2.9. Mathematics Tutorial 26

3. Probability 30

3.1. Measurement 30

3.2. Randomness 32

3.3. A Standard for Probability 34

3.4. Probability 35

3.5. Coherence 36

3.6. Belief 37

3.7. Complementary Event 39

3.8. Odds 40

3.9. Knowledge Base 43

3.10. Examples 44

3.11. Retrospect 46

4. Two Events 47

4.1. Two Events 47

4.2. Conditional Probability 49

4.3. Independence 51

4.4. Association 53

4.5. Examples 54

4.6. Supposition and Fact 56

4.7. Seeing and Doing 57

5. The Rules of Probability 59

5.1. Combinations of Events 59

5.3. Multiplication Rule 62

5.4. The Basic Rules 64

5.5. Examples 66

5.6. Extension of the Conversation 68

5.7. Dutch Books 70

5.8. Scoring Rules 72

5.9. Logic Again 73

5.10. Decision Analysis 74

5.11. The Prisoners’ Dilemma 75

5.12. The Calculus and Reality 76

6. Bayes Rule 79

6.1. Transposed Conditionals 79

6.2. Learning 81

6.3. Bayes Rule 82

6.4. Medical Diagnosis 83

6.5. Odds Form of Bayes Rule 86

6.6. Forensic Evidence 88

6.7. Likelihood Ratio 89

6.8. Cromwell’s Rule 90

6.9. A Tale of Two Urns 92

6.10. Ravens 94

6.11. Diagnosis and Related Matters 97

6.12. Information 98

7. Measuring Uncertainty 101

7.1. Classical Form 101

7.2. Frequency Data 103

7.3. Exchangeability 104

7.4. Bernoulli Series 106

7.5. De Finetti’s Result 107

7.6. Large Numbers 109

7.7. Belief and Frequency 111

7.8. Chance 114

8. Three Events 117

8.1. The Rules of Probability 117

8.3. Source of the Paradox 121

8.4. Experimentation 122

8.5. Randomization 123

8.6. Exchangeability 125

8.7. Spurious Association 128

8.8. Independence 130

8.9. Conclusions 132

9. Variation 134

9.1. Variation and Uncertainty 134

9.2. Binomial Distribution 135

9.3. Expectation 137

9.4. Poisson Distribution 139

9.6. Variability as an Experimental Tool 144

9.7. Probability and Chance 145

9.8. Pictorial Representation 147

9.9. The Normal Distribution 150

9.10. Variation as a Natural Phenomenon 152

10. Decision Analysis 158

10.1. Beliefs and Actions 158

10.2. Comparison of Consequences 160

10.3. Medical Example 162

10.4. Maximization of Expected Utility 164

10.5. More on Utility 165

10.6. Some Complications 167

10.7. Reason and Emotion 168

10.8. Numeracy 170

10.9. Expected Utility 171

10.10. Decision Trees 172

10.11. The Art and Science of Decision Analysis 175

10.12. Further Complications 177

10.13. Combination of Features 179

10.14. Legal Applications 182

11. Science 186

11.1. Scientific Method 186

11.2. Science and Education 187

11.3. Data Uncertainty 188

11.4. Theories 190

11.5. Uncertainty of a Theory 193

11.6. The Bayesian Development 195

11.7. Modification of Theories 197

11.8. Models 199

11.9. Hypothesis Testing 202

11.10. Significance Tests 204

11.11. Repetition 206

11.12. Summary 208

12. Examples 211

12.1. Introduction 211

12.2. Cards 212

12.3. The Three Doors 213

12.4. The Newcomers to Your Street 215

12.5. The Two Envelopes 217

12.6. Y2K 220

12.7. UFOs 221

12.8. Conglomerability 224

13. Probability Assessment 226

13.1. Nonrepeatable Events 226

13.2. Two Events 227

13.3. Coherence 230

13.4. Probabilistic Reasoning 233

13.5. Trickle Down 234

13.6. Summary 236

Epilogue 238

Subject Index 243

Index of Examples 248

Index of Notations 250

## Books & Journals

### Books #### Log-Linear Modeling: Concepts, Interpretation, and Application #### Bayesian Estimation and Tracking: A Practical Guide View all

### Journals #### WIREs Computational Statistics #### Significance View all